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Question Number 157016 Answers: 1 Comments: 0

f(x)=(1/(1+2^x ))+(1/(1+3^x ))+(1/(4^x +1)) find ∫_1 ^5 f(x)dx+∫_(−5) ^(−1) f(x)dx

$${f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}^{{x}} }+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{3}^{{x}} }+\frac{\mathrm{1}}{\mathrm{4}^{{x}} +\mathrm{1}}\:\: \\ $$$${find}\:\:\:\int_{\mathrm{1}} ^{\mathrm{5}} {f}\left({x}\right){dx}+\int_{−\mathrm{5}} ^{−\mathrm{1}} {f}\left({x}\right){dx} \\ $$

Question Number 157010 Answers: 0 Comments: 3

Question Number 157007 Answers: 1 Comments: 1

Question Number 157006 Answers: 1 Comments: 0

(√(4+27(√(4+29(√(4+31(√(4+…))))))))=?

$$\sqrt{\mathrm{4}+\mathrm{27}\sqrt{\mathrm{4}+\mathrm{29}\sqrt{\mathrm{4}+\mathrm{31}\sqrt{\mathrm{4}+\ldots}}}}=? \\ $$$$ \\ $$

Question Number 156992 Answers: 1 Comments: 1

Question Number 156991 Answers: 1 Comments: 0

Question Number 156979 Answers: 2 Comments: 2

Question Number 156977 Answers: 0 Comments: 0

{ ((a_1 =((√3)/2))),((a_(n+1) =4a_n ^3 −3a_n ; ∀n≥1)) :} a_n =?

$$\:\begin{cases}{{a}_{\mathrm{1}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\\{{a}_{{n}+\mathrm{1}} =\mathrm{4}{a}_{{n}} ^{\mathrm{3}} −\mathrm{3}{a}_{{n}} \:;\:\forall{n}\geqslant\mathrm{1}}\end{cases} \\ $$$$\:{a}_{{n}} =? \\ $$

Question Number 156973 Answers: 0 Comments: 1

Question Number 156993 Answers: 0 Comments: 3

lim_(n→∞) (sin(sin(sin…(sin(x))…)_(n) (√n)=? 0<x<π

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{n}} {\underbrace{\left({sin}\left({sin}\left({sin}\ldots\left({sin}\left({x}\right)\right)\ldots\right)}}\:\sqrt{{n}}=?\right.\right. \\ $$$$\mathrm{0}<{x}<\pi \\ $$

Question Number 157003 Answers: 1 Comments: 0

let n∈Z^+ shov that ∫_( 0) ^( ∞) ((sin(x^(-n) )ln(x))/x) dx = ((π𝛄)/(2n^2 )) where 𝛄 is the Euler-Mascheroni constan

$$\mathrm{let}\:\:\boldsymbol{\mathrm{n}}\in\mathbb{Z}^{+} \\ $$$$\mathrm{shov}\:\mathrm{that}\:\:\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{sin}\left(\mathrm{x}^{-\boldsymbol{\mathrm{n}}} \right)\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{dx}\:=\:\frac{\pi\boldsymbol{\gamma}}{\mathrm{2n}^{\mathrm{2}} }\: \\ $$$$\mathrm{where}\:\:\boldsymbol{\gamma}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{Euler}-\mathrm{Mascheroni}\:\mathrm{constan}\: \\ $$

Question Number 156969 Answers: 0 Comments: 0

Question Number 156968 Answers: 0 Comments: 0

Question Number 156966 Answers: 2 Comments: 1

Question Number 156962 Answers: 1 Comments: 0

Solve for real numbers: (3/( ((1 + x))^(1/3) )) + (x/( ((1 + x^3 ))^(1/3) )) = 2 (4)^(1/3)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{3}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}\:+\:\mathrm{x}}}\:+\:\frac{\mathrm{x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }}\:=\:\mathrm{2}\:\sqrt[{\mathrm{3}}]{\mathrm{4}} \\ $$

Question Number 156961 Answers: 0 Comments: 0

𝛀 =∫_( 0) ^( ∞) ((cos^2 (x) - sin^2 (x))/((1 + x^4 )^3 )) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)\:-\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{3}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 156951 Answers: 1 Comments: 7

solve for n∈N (n−1)!+1=n^2

$${solve}\:{for}\:{n}\in{N} \\ $$$$\left({n}−\mathrm{1}\right)!+\mathrm{1}={n}^{\mathrm{2}} \\ $$

Question Number 156942 Answers: 0 Comments: 0

Question Number 156940 Answers: 1 Comments: 0

Question Number 156933 Answers: 1 Comments: 0

Question Number 156931 Answers: 1 Comments: 2

(1−x^2 )y′′−4xy′−(1+x^2 )y=0

$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}''−\mathrm{4}{xy}'−\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}=\mathrm{0} \\ $$

Question Number 156927 Answers: 0 Comments: 0

Find: 𝛀 =lim_(n→∞) ∫_( 0) ^( 5) (((1 - x)∙x^(n+4) )/(1 + x^(3n) )) dx = ?

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{\left(\mathrm{1}\:-\:\mathrm{x}\right)\centerdot\mathrm{x}^{\boldsymbol{\mathrm{n}}+\mathrm{4}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}\boldsymbol{\mathrm{n}}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 156924 Answers: 1 Comments: 0

Π_(k=2) ^∞ (((k^3 −1)/(k^3 +1))) =?

$$\:\underset{{k}=\mathrm{2}} {\overset{\infty} {\prod}}\:\left(\frac{{k}^{\mathrm{3}} −\mathrm{1}}{{k}^{\mathrm{3}} +\mathrm{1}}\right)\:=? \\ $$

Question Number 156923 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( (𝛑/4)) x log (1 + tanx) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\mathrm{x}\:\mathrm{log}\:\left(\mathrm{1}\:+\:\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 156921 Answers: 1 Comments: 0

x^3 −4x^2 −3=0 x∈R

$${x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0} \\ $$$$\:{x}\in\mathbb{R} \\ $$

Question Number 156913 Answers: 1 Comments: 0

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